Nuprl Lemma : harmonic-series-diverges-to-infinity
∀n:ℕ. ((r1 + (r(n)/r(2))) ≤ Σ{(r1/r(i)) | 1≤i≤2^n})
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
rdiv: (x/y),
rleq: x ≤ y,
radd: a + b,
int-to-real: r(n),
exp: i^n,
nat: ℕ,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
so_apply: x[s],
subtype_rel: A ⊆r B,
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
rge: x ≥ y,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
less_than'_wf,
rsub_wf,
rsum_wf,
exp_wf2,
nat_plus_properties,
rdiv_wf,
int-to-real_wf,
int_seg_properties,
int_seg_wf,
radd_wf,
nat_plus_wf,
le_wf,
rless-int,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
rless_wf,
decidable__le,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
nat_wf,
rsum-single,
req_weakening,
rleq_functionality,
int_formula_prop_eq_lemma,
intformeq_wf,
int_subtype_base,
equal-wf-base,
exp0_lemma,
radd-int,
rdiv-zero,
radd_functionality,
uiff_transitivity,
false_wf,
rleq-int-fractions2,
rleq_wf,
exp_step,
rsum-split,
exp_wf_nat_plus,
le_weakening2,
itermMultiply_wf,
int_term_value_mul_lemma,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rsum_functionality_wrt_rleq,
rleq-int-fractions,
mul_nat_plus,
rmul_wf,
rneq-int,
int_entire_a,
subtype_base_sq,
true_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
rsum-constant2,
decidable__equal_int,
rmul_preserves_req,
rinv_wf2,
rneq_functionality,
rmul-int,
req_functionality,
req_transitivity,
rmul_functionality,
rinv_functionality2,
req_inversion,
rinv-of-rmul,
real_term_polynomial,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul-rinv,
rmul-rinv3,
rmul-rdiv-cancel,
radd_comm,
radd-assoc,
rmul_comm,
rmul-one-both,
rmul-distrib,
rleq-int,
rmul_preserves_rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
productElimination,
independent_pairEquality,
applyEquality,
because_Cache,
addEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
inrFormation,
unionElimination,
imageMemberEquality,
baseClosed,
setEquality,
multiplyEquality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
addLevel,
instantiate,
cumulativity,
equalityElimination,
impliesFunctionality,
functionEquality
Latex:
\mforall{}n:\mBbbN{}. ((r1 + (r(n)/r(2))) \mleq{} \mSigma{}\{(r1/r(i)) | 1\mleq{}i\mleq{}2\^{}n\})
Date html generated:
2017_10_03-AM-09_19_40
Last ObjectModification:
2017_07_28-AM-07_44_32
Theory : reals
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